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Hilbert’s twentyfourth problem
 American Mathematical Monthly
, 2001
"... 1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Cong ..."
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1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of
COHEN AND SET THEORY
"... Abstract. We discuss the work of Paul Cohen in set theory and its influence, especially the background, discovery, development of forcing. Paul Joseph Cohen (1934–2007) in 1963 established the independence of the Axiom of Choice (AC) from ZF and the independence of the Continuum Hypothesis (CH) from ..."
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Abstract. We discuss the work of Paul Cohen in set theory and its influence, especially the background, discovery, development of forcing. Paul Joseph Cohen (1934–2007) in 1963 established the independence of the Axiom of Choice (AC) from ZF and the independence of the Continuum Hypothesis (CH) from ZFC. That is, he established that Con(ZF) implies Con(ZF+¬AC) and Con(ZFC) implies Con(ZFC+¬CH). Already prominent as an analyst, Cohen had ventured into set theory with fresh eyes and an openmindedness about possibilities. These results delimited ZF and ZFC in terms of the two fundamental issues at the beginnings of set theory. But beyond that, Cohen’s proofs were the inaugural examples of a new technique, forcing, which was to become a remarkably general and flexible method for extending models of set theory. Forcing has strong intuitive underpinnings and reinforces the notion of set as given by the firstorder ZF axioms with conspicuous uses of Replacement and Foundation. If Gödel’s construction of L had launched set theory as a distinctive field of mathematics, then Cohen’s forcing began its transformation into a modern, sophisticated one. The extent and breadth of the expansion of set theory henceforth dwarfed all that came before, both in terms of the numbers of people involved and the results established. With clear intimations of a new and concrete way of building models, set theorists rushed in and with forcing were soon establishing a cornucopia of relative consistency results, truths in a wider sense, with some illuminating classical problems of mathematics. Soon, ZFC became quite unlike Euclidean geometry and much like group theory, with a wide range of models of set theory being investigated for their own sake. Set theory had undergone a seachange, and with the subject so enriched, it is difficult to convey the strangeness of it. Received April 24, 2008. This is the full text of an invited address given at the annual meeting of the Association
Abstract—This is a brief overview of the basic techniques of Boolean valued analysis.
, 2006
"... The term “Boolean valued analysis ” appeared within the realm of mathematical logic. It was Takeuti, a renowned expert in proof theory, who introduced the term. Takeuti defined Boolean valued analysis in [18, p. 1] as “an application of Scott–Solovay’s Boolean valued models of set theory to analysis ..."
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The term “Boolean valued analysis ” appeared within the realm of mathematical logic. It was Takeuti, a renowned expert in proof theory, who introduced the term. Takeuti defined Boolean valued analysis in [18, p. 1] as “an application of Scott–Solovay’s Boolean valued models of set theory to analysis.” Vopěnka invented similar models at the same time. That is how the question of the title receives
Chapter 2 Fourier Series and Integrals
"... Having contested the various results [Biot and Poisson] now recognise that they are exact but they protest that they have invented another method of expounding them and that this method is excellent and the true one. If they had illuminated this branch of physics by important and general views and h ..."
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Having contested the various results [Biot and Poisson] now recognise that they are exact but they protest that they have invented another method of expounding them and that this method is excellent and the true one. If they had illuminated this branch of physics by important and general views and had greatly perfected the analysis of partial differential equations, if they had established a principal element of the theory of heat by fine experiments... they would have the right to judge my work and to correct it. I would submit with much pleasure... But one does not extend the bounds of science by presenting, in a form said to be different, results which one has not found oneself and, above all, by forestalling the true author in publication.
WHAT IS BOOLEAN VALUED ANALYSIS?
, 2006
"... Abstract. This is a brief overview of the basic techniques of Boolean valued analysis. 1. ..."
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Abstract. This is a brief overview of the basic techniques of Boolean valued analysis. 1.
BOOK REVIEWS BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
"... The literature of mathematics comprises millions of works, published ones as well as ones deposited in electronic archives. The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thous ..."
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The literature of mathematics comprises millions of works, published ones as well as ones deposited in electronic archives. The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year [28]. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. In addition, many works also formulate unsolved problems, often in the form of precise conjectures. How essential is it for the development of mathematical science to draw the readers’ attention unceasingly to open problems? Maybe it would suffice to publish only new results? The firstrank mathematicians of the present time give a definitive answer to this question. In his preface to the first Russian edition [20] of the book under review, 1 V. I. Arnold reminisced: “I. G. Petrovskiĭ, who was one of my teachers in Mathematics, taught me that the most important thing that a student should learn from his supervisor is that some question is still open. Further choice of the problem from